scholarly journals A method for solving a boundary value problem in a multilayered area

2020 ◽  
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Boundary Value ◽  
Internal Heat ◽  
Electrical Machine ◽  
Problem Solution ◽  
Conductive Heat ◽  
Heat Sources ◽  
Internal Layers ◽  
Internal Heat Sources

A mathematical model of thermal process in an electrical machine was built as an example, presented as a three-layer cylinder where internal heat sources operate in one of the layers and heat is submitted to the other two by means of heat conduction. A method of solving the boundary-value problems for heat conduction equation in a complex area – a multi-layered cylinder with internal heat sources operating in one part of the layers and external ones in another part, is proposed. A method of problem solution in conditions of uncertainty of one of the boundary condition at the layers interface with conductive heat exchange between the layers is reviewed. The principle of method lies in the averaging of temperature distributions radially in the internal layers. As a result of transformations at the layers interface a boundary condition of the impedance-type conjugation appears. The analytical and numeric-analytical solutions of simplified problems were obtained.

scholarly journals Thermal constitutive matrix applied to asynchronous electrical machine using the cell method

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 27-30 ◽  
Author(s):  
Pablo Ignacio González Domínguez ◽  
José Miguel Monzón-Verona ◽  
Leopoldo Simón Rodríguez ◽  
Adrián de Pablo Sánchez
Keyword(s):  
Heat Conduction ◽  
Internal Heat ◽  
Conduction Equation ◽  
Equivalent Model ◽  
Electrical Machine ◽  
Heat Sources ◽  
Conduction Model ◽  
Cell Method ◽  
Internal Heat Sources ◽  
Constitutive Matrix

Abstract This work demonstrates the equivalence of two constitutive equations. One is used in Fourier’s law of the heat conduction equation, the other in electric conduction equation; both are based on the numerical Cell Method, using the Finite Formulation (FF-CM). A 3-D pure heat conduction model is proposed. The temperatures are in steady state and there are no internal heat sources. The obtained results are compared with an equivalent model developed using the Finite Elements Method (FEM). The particular case of 2-D was also studied. The errors produced are not significant at less than 0.2%. The number of nodes is the number of the unknowns and equations to resolve. There is no significant gain in precision with increasing density of the mesh.

A New Uniform Continuum Modeling of Conductive and Radiative Heat Transfer in Nuclear Pebble Bed

2019 ◽  
Vol 141 (8) ◽  
Author(s):  
Hao Wu ◽  
Nan Gui ◽  
Xingtuan Yang ◽  
Jiyuan Tu ◽  
Shengyao Jiang
Keyword(s):  
Heat Transfer ◽  
Continuum Model ◽  
Internal Heat ◽  
Size Factor ◽  
Conductive Heat ◽  
Heat Sources ◽  
Pebble Bed ◽  
Internal Heat Sources ◽  
Good Agreement ◽  
The Continuum

Radiative and conductive heat transfer is fairly important in the nuclear pebble bed. A continuum model is proposed here to derive the effective thermal conductivity (ETC) of pebble bed. It is a physics-based equation determined by the temperature, number density, heat transfer coefficient, and the radial distribution function (RDF). Based on a concept of continuum, this model considers the conduction and thermal radiation in nuclear pebble bed through a uniform framework and the results are in good agreement with the existing model and correlations. It indicates that the local temperature in the radiation case without internal heat sources is determined by all possible surrounding pebbles weighted by a radiative kernel function. The discrete element method (DEM) packing results are in good agreement with the solution of the continuum model. Both the conductive and radiative continuum models converge to the heat conduction in continuum mechanics at size factor μ ≪ 1.

Boundary Condition Dependent Natural Convection in a Rectangular Pool With Internal Heat Sources

2006 ◽  
Vol 129 (5) ◽  
pp. 679-682 ◽  
Author(s):  
Seung Dong Lee ◽  
Jong Kuk Lee ◽  
Kune Y. Suh
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Natural Convection ◽  
Convection Heat Transfer ◽  
Internal Heat ◽  
Working Fluid ◽  
Natural Convection Heat Transfer ◽  
Heat Sources ◽  
Internal Heat Sources ◽  
Volumetric Heat Generation

This paper presents results of steady-state experiments concerned with natural convection heat transfer of air in a rectangular pool in terms of the Nusselt number (Nu) versus the modified Rayleigh number (Ra′) varying from 109 to 1012. Cartridge heaters were immersed in the working fluid to simulate uniform volumetric heat generation. Two types of boundary conditions were adopted in the test: (I) top cooled, and (II) top and bottom cooled. The other sides were kept insulated. In the case of boundary condition II, the upward heat transfer ratio, Nuup∕(Nuup+Nudn), turned out to be 0.7–0.8 in the range of Ra′ between 1.05×1010 and 3.68×1011.

An inverse heat conduction boundary problem for a two-part rod with different thermal conductivity

2021 ◽  
Vol 9 (1) ◽  
pp. 69-86
Author(s):  
V.P. Tanana ◽  
◽  
A.I. Sidikova ◽  
B.A. Markov ◽  
◽  
...  
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Boundary Value Problem ◽  
Direct Problem ◽  
Boundary Value ◽  
Dirichlet Condition ◽  
Problem Solution ◽  
Inverse Boundary ◽  
Inverse Boundary Value Problem ◽  
The Right

The paper studies the problem of determining the boundary condition in the heat conduction equation for a rod consisting of homogeneous parts with different thermophys- ical properties. We consider the Dirichlet condition at the left end of the rod (at x = 0) corresponding to the heating of this end and the homogeneous condition of the first kind at the right end (at x = 1) corresponding to cooling during interaction with the environment as boundary conditions. At the point of discontinuity of the thermophysical properties (at x = x0), the conditions for the continuity of temperature and heat flux are set. In the inverse problem, the boundary condition at the left end is assumed to be unknown. To find it, the value of the direct problem solution at the point x0, i.e., the point of separation of the rod into two homogeneous sections, is set. In this work, we carried out an analytical study of the direct problem, which allowed us to apply the time Fourier transform to the inverse boundary value problem. The projection-regularization method is used to solve the inverse boundary value problem for the heat equation and obtain error estimates of this solution correct to the order.

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